Method for calculating parameters in road design of S-type clothoid, complex clothoid and egg type clothoid

ABSTRACT

The present invention relates to a method for calculating parameters in a road design of a S type, a complex type and an egg type clothoid, and in particular to a method for calculating a parameter value capable of determining the size of a clothoid that is inserted when designing a S-shaped and interchange, a connection road, etc. in an egg shape. In the present invention, it is possible to easily calculate the clothoid parameter A in a S shape, complex type and egg type road design, and a road design can be fast finished. In addition, in the present invention, it is possible to achieve an easier design of a S shape and egg type clothoid by determining a design specification without using CAD. The design can be achieved based on a simulation using a center coordinate of two circles for achieving an optimum design.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for calculating parameters ina road design of a S type, a complex type and an egg type clothoid, andin particular to a method for calculating a parameter value capable ofdetermining the size of a clothoid that is inserted when designing aS-shaped and interchange, a connection road, etc. in an egg shape.

2. Description of the Background Art

Generally, when a vehicle parks at a road directly connecting a straightline and a circle, the vehicle receives a sharp centrifugal accelerationor a rotational angular speed when the radius of a circle is small, sothat passenger feels uncomfortable or there is something dangerous indriving. Therefore, a smooth curve may be inserted between theconnections for thereby decreasing the above problems. The used in thepresent invention are defined as follows.

“Clothoid” represents a curve of which a curvature (reverse number ofradius) is increased in proportion to the length of a curve and adriving trace that a vehicle makes when the vehicle runs at a constantspeed, and the rotation angular speed of the front wheels is constant.

The following equations are obtained at all points on one clothoid.

(curvature radius R at a certain point)×(curve length L from the centerof the clothoid to the point)=(constant value A²). The above equation(namely, R×L=A² is called the basic formula of the clothoid. Allelements of the clothoid are induced based on the above basic formula.

Here, the clothoid may be classified into a basic type (a connection ina sequence of straight line, clothoid, circular curve, clothoid, andstraight line), a S shape (two clothoid are inserted between reflectioncurves), an egg type (clothoid is inserted between double centercurves), a protrusion type (two clothoid bent in the same direction areconnected with each other), and a complex type (at least two clothoidbent in the same direction). The basic type has been basically usedduring the design. The basic type design can be easily achieved usingthe clothoid formulas). The complex type has not used yet. Theinterchange and connection road are designed in the egg type. Thecalculation methods of the S type, complex type and egg type aredifficult. It is impossible to easily calculate with only the basisformula.

In the S type and egg type, the important thing is to calculate thevalue of the parameter A determining the size of the clothoid inserted.It is impossible to easily calculate with only the basic formula of theclothoid differently from the basic type.

The egg type has been generally used for the interchange or theconnection road. The interchanges are actually used in a combination ofone or at least two egg types. The egg type has been generally used inthe interchanges of a straight connection type, clover type and trumpettype. Each type can be combined in the independent egg type. The forwarddirection egg type is an egg type that the linearity is formed in thedirections of the entrance and exist axis crossing points. FIG. 1 is aview illustrating the type of a forward direction egg type.

The backward direction egg type is a type that the linearity is formedin the direction of the entrance and exist axis crossing points. FIG. 2is a view of the backward direction egg type.

The S shape egg type is an egg type that a smoothing curve is installedbetween the short curves bent in the opposite direction. FIG. 3 is aview of the egg type.

The S type clothoid is the type that the clothoid curve is installed sothat the viewing times are same with respect to two circles positionedin the direction opposite to the common axis. Here, the parameterrepresents the value A of the parameter of the clothoid curve.

The double egg type is the type that the connection is made using twoegg types based on the assistant circle in the case that two circles arecrossed or are distanced. The double egg type can be classified intofour types as shown in FIG. 4. Namely, there are (i) the type that usesthe assistant circle including two crossing circles, (ii) the type thatuses the assistant circle having two distanced circles, (iii) the typethat uses the assistant circle included in two crossing circles and (iv)the type that uses the assistant circle used because the distance in theradiuses of two circles is too large.

The conventional design method of the S type and egg type roads will bedescribed.

[Conventional Design Method of S Shape Clothoid]

When the S shape clothoid curve is designed, the smoothing curve isinstalled with respect to the common axis so that a straight line doesnot exist between the smoothing curves of two circles. The parametervalues A with respect to the circles 1 and 2 are generally set with thesame values, but may be different in some special cases based on thedesign characteristic. However, there is not any formula for accuratelycalculating the parameters with respect to the S type clothoid.Therefore, the S type clothoid can be not designed at one time, so thatit is separately designed by classifying the circles 1 and 2. Namely,the circle 1 is designed with a symmetric type or a non-symmetric type.The design specification with respect to the circle 2 is designed sothat a straight line is not formed between the circles 1 and 2 using aresult of the design of the circle 1. However, in the conventionalmethod, it is impossible to set the accurate specification, and it takeslong time. Many tests should be performed until a desired result isobtained.

[Conventional Method of Egg Type Design]

The important thing of the egg type design is to determine the parameterA with respect to the smoothing curve installed between a larger circleand a smaller circle. However, there is not currently any formula foraccurately calculating the parameter value A. In addition, it isimpossible to accurately design and calculate each program. In thecurrently available programs, the design is performed at one time in thecase that a designer designates the parameter value A like the radius ofthe circle. However, it is actually impossible to calculate theparameter value A for a desired design. Therefore, a proper value isdesignated and designed. The above process is repeated until a desireddesign is achieved.

Generally, the value A is not integer and should be calculated down tofour˜six decimal places in order to use an accurate value in apermissible value in the actual work. Therefore, much efforts should beprovided in order to determine the value within a permissible error inthe actual work. Since the egg type is mainly used in the interchange orthe connection road, the coordinates of the entrance and exist axis atwhich the egg type is installed has been already determined. Therefore,the start point of the egg type should be provided at the axis of theentrance as a result of the egg type design when designing the egg type,and the ending point should be provided on the axis of the exist, sothat the linearities of the front and rear sides of the egg type are notchanged. The egg type is not designed at a desired position unless theparameter value A of the egg type is designated with an accurate valuewithin the permissible error. The linearity portions after the egg typeget changed as compared to the set linearity.

The egg type generally uses a single egg type. In a special case basedon the design characteristic, the double egg type may be used. In thedouble egg type, there is not any accurate design method.

As the prior art related to the present invention, there is an appliedmeasurement (written by Hong Hyun Ki and published by Seoul IndustrialUniversity and published n Feb. 26, 1993) in which the basic formulasand assistant materials for design with respect to the S shape clothoid,egg type clothoid and complex type clothoid design calculations aredisclosed. The above theories are obtained based on the books of DIEKLOTOIDE als TRANSSIERUNGSELEMENT published in 1964 in Germany andSTRASENPLANUNG mit KLOTOIDEN written by Horst Osterloh and published in1965.

The value A in the egg type can be obtained using the diagram of HorstOsterloh and the table made by Kasper, Schuerba and Lorenz. The diagramof Osterlok can not be actually used in the actual work, and the tablemade by Kasper, Schuerba, Lorenz can be actually used in the actual workand can be applied in the programs. In this case, the table is used, sothat the size of the program is increased, and the values of the tablesare informal values, and multiple tables should be used. The values areobtained by approximate values. In the case that the values are out of acertain range, there may be a big difference from the actual value.

SUMMARY OF THE INVENTION

Accordingly, it is an object of the present invention to overcome theproblems encountered in the conventional art.

It is another object of the present invention to provide a method forcalculating a parameter value adapted to determine the size of aclothoid when the egg type design is performed in the S shape, complextype, interchange and connection road based on the design condition ofthe roads. To achieve the above object, there is provided a method forcalculating a clothoid parameter value A of the S type and egg typebased on a mathematical method through a geometrical interpretationmethod. Namely, the formula for calculating the parameter value A of theS type and egg type and inducing process of the formula will bedescribed.

It is further another object of the present invention to provide amethod for calculating the roads of S type, complex type and egg type insuch a manner that a specification needed for the design isautomatically calculated without CD as compared to the conventional artin which necessary specifications needed for the design are alldesignated, and each element is continuously drawn based on a result ofthe designation.

It is still further another object of the present invention to provide amethod for calculating a clothoid parameter in a road design of S type,complex type and egg type that can be simulated in various methods forselecting the coordinate of the center point for the optimum design.

To achieve the above objects, in a method for calculating a S shapeclothoid parameter including a unknown clothoid parameter A adapted toradiuses (R₁R₂) of two circles, the shortest distance betweencircumferential portions of two circles and two circles, there isprovided a method for calculating a S shape clothoid parameter,comprising a step in which an initial value of a tangential angle τ₁, isset; a step in which the value of (R₁+D+R₂)²−Xm² is calculated in such amanner that the tangential angle (τ₁) is compared, and when a result ofthe comparison is below 0°, since it means there is not any resolution,the process is stopped, and when a result of the same is over 0°, theprocess is continued; a step in which the value of (R₁+D+R₂)2−Xm² iscompared with 0, and when a result of the comparison is below 0, thetangential angle is properly adjusted, and the routine goes back to thestep for setting the initial value of the tangential value of (τ₁), andwhen a result of the same is over 0, a different fixed rate formula isset up with respect to two circles, and one formula is formed by addingthe left and right items in two formulas, for thereby obtaining atangential angle (τ₁); a step in which the function F (τ₁) of thetangential angle (τ₁) and the function are differentiated with thetangential angle (τ₁) for thereby calculating the differential functionF′(τ₁); a step in which the ratio [G=(F(τ₁)/F′(τ₁)] of two functions of[(F(τ₁), F′(τ₁)] are calculated; and a step in which the absolute valueof the ratio(G) is compared with a permissible error (10⁻⁶), and as aresult of the comparison when it is over the permissible error, theinitial value of the tangential value (τ₁) is set, and the routine isfed back to the next step of the step that the initial value is set, andwhen it is below the permissible error, the tangential angle (τ₁) isdetermined, and the parameter value A is calculated using the tangentialangle (τ₁).

In the step for calculating the tangential angle with respect to theradius of the circle, the smoothing curve length is obtained in stead ofobtaining the tangential angle with respect to the radius of the circlefor the reasons that it is A²=R*L=2τR², so that the case for calculatingthe tangential angle (τ) and the case for obtaining the smoothing curvelength (L) are same variables for obtaining the parameters.

To achieve the above objects, in a method for calculating a S shapeclothoid parameter including a unknown clothoid parameter A adapted toradius (R₁ R₂) of two circles, the shortest distance betweencircumferential portions of two circles and two circles, there isprovided a method for calculating an egg type clothoid parameter,comprising a step in which an initial value of a tangential angle T₁ isset; a step in which the value of (R₁−R₂−D)²−Xm² is calculated in such amanner that the tangential angle (τ₁) is compared, and when a result ofthe comparison is below 0°, since it means there is not any resolution,the process is stopped, and when a result of the same is over 0°, theprocess is continued; a step in which the value of (R₁–R₂−D)²−Xm² iscompared with 0, and when a result of the comparison is below 0, thetangential angle is properly adjusted, and the routine goes back to thestep for setting the initial value of the tangential value of (τ₁), andwhen a result of the same is over 0, a different fixed rate formula isset up with respect to two circles, and one formula is formed by addingthe left and right items in two formulas, for thereby calculating F′(τ₁)by differentiating the function F(τ₁) of the tangential angle(τ₁) andthe function with the tangential angle (τ₁); a step in which the ratioof [G=(F(τ₁)/F′(τ₁)] of two functions of [(F(τ₁), F′(τ₁)] is calculated;a step in which the tangential angle of τ₁=τ₁−G is calculated; and astep in which the absolute value of the ratio(G) is compared with apermissible error (10 ⁻⁶), and as a result of the comparison when it isover the permissible error, the initial value of the tangential value(τ₁) is set, and the routine is fed back to the next step of the stepthat the initial value is set, and when it is below the permissibleerror, the tangential angle (τ₁) is determined, and the parameter valueA is calculated using the tangential angle (τ₁).

In addition, the clothoid parameter is calculated with respect to themultiple egg type (egg type of more than double egg type) using the eggtype road clothoid parameter calculation method. The double egg type isrecognized as an independent egg type, and the egg type clothoidparameter value is calculated with respect to each independent egg type.

The double egg type road designing method can be selected from the groupcomprising a method that designs using an assistant circle including twocrossing circles; a method that designs using an assistant circleincluding two distanced circles; a method that designs using anassistant circle included in the two crossing circles; and a method thatdesigns using an assistant circle including a smaller circle andincluded in a larger circle.

Here, the egg type is the type in which there is one egg type clothoidin two circles. In this case, the smaller circle should be includedwithin the larger circle. In a special case, the double egg type can beinstalled using three circles. In this case, there are two egg typeclothoid. The above construction can be directly adapted to the doubleegg type. Namely, the formula for calculating the parameter A withrespect to the egg type clothoid can be directly adapted whencalculating the parameter A with respect to the double egg type or thetriple egg type.

Here, the double egg type is the type in which two egg types arecontinuously arranged. When designing the double egg type, it is neededto independently design two continuous egg types. Therefore, when thecalculation is performed using the design method of the egg type, thedouble egg type design can be achieved. The above method can be adaptedto the multiple egg type of over double egg types.

The complex type clothoid is very similar with the egg type except forthe differences that the circular portion with respect to the largercircle exists in the egg type, but does not exist in the complex type.(The length of the circle and the center angle of the circle are all 0).The method for calculating the parameter A of the complex clothoid issame as the method for calculating the parameter A in the egg typeclothoid. When the parameter A is obtained, it is possible to easilydesign the complex clothoid. The design itself of the complex typeclothoid can directly adapt the method of the egg type design. However,it is needed to process the center angle of the circle with respect tothe larger circle to 0. Therefore, since the method for calculating theparameter A of the egg type clothoid in the present invention can bedirectly adapted to the method of calculating the parameter A in thecomplex type clothoid.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will become better understood with reference tothe accompanying drawings which are given only by way of illustrationand thus are not limitative of the present invention, wherein

FIG. 1 is a view illustrating a forward direction egg type;

FIG. 2 is a view illustrating a backward direction egg type;

FIG. 3 is a view illustrating a S-shaped egg type;

FIG. 4 is a view illustrating a double egg type;

FIG. 5 is a view illustrating a S type clothoid according to the presentinvention;

FIG. 6 is a view illustrating a egg type clothoid according to thepresent invention;

FIG. 7 is a flow chart illustrating a S type clothoid parametercalculation method according to the preset invention;

FIG. 8 is a flow chart illustrating an egg type clothoid parametercalculation method according to the present invention;

FIG. 9 is a view illustrating a forward direction egg type designaccording to the present invention;

FIG. 10 is a view illustrating a backward direction egg type designaccording to the present invention;

FIG. 11 is a view illustrating a S-shaped egg type design according tothe present invention;

FIG. 12 is a view illustrating a double egg type using an assistantcircle having two crossing circles according to the present invention;

FIG. 13 is a view illustrating a double egg type using an assistantcircle having two distanced circles according to the present invention;

FIG. 14 is a view illustrating a double egg type using an assistantcircuit having two crossing circles according to the present invention;

FIG. 15 is a view illustrating a double egg type using an assistantcircle because a distance between the distances in the radiuses of twocircles according to the present invention;

FIG. 16 is a view illustrating a design example of a triple egg typeaccording to the present invention;

FIG. 17 is a view illustrating a complex type clothoid constructionaccording to the present invention; and

FIG. 18 is a view illustrating a complex type clothoid according to thepresent invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The preferred embodiments of the present invention will be describedwith reference to the accompanying drawings.

FIG. 5 is a view illustrating a S type clothoid according to the presentinvention, FIG. 6 is a view illustrating a egg type clothoid accordingto the present invention, FIG. 7 is a flow chart illustrating a S typeclothoid parameter calculation method according to the preset invention,and FIG. 8 is a flow chart illustrating an egg type clothoid parametercalculation method according to the present invention.

FIG. 9 is a view illustrating a forward direction egg type designaccording to the present invention, FIG. 10 is a view illustrating abackward direction egg type design according to the present invention,and FIG. 11 is a view illustrating a S-shaped egg type design accordingto the present invention.

FIG. 12 is a view illustrating a double egg type using an assistantcircle having two crossing circles according to the present invention,

FIG. 13 is a view illustrating a double egg type using an assistantcircle having two distanced circles according to the present invention,FIG. 14 is a view illustrating a double egg type using an assistantcircuit having two crossing circles according to the present invention,FIG. 15 is a view illustrating a double egg type using an assistantcircle because a distance between the distances in the radiuses of twocircles according to the present invention, and FIG. 16 is a viewillustrating a design example of a triple egg type according to thepresent invention;

First Embodiment Calculation Method of A₁, A₂ at S Type Clothoid

1. Characteristic of S Type Clothoid

As shown in FIG. 5, in the S type clothoid, two circles having radius ofR₁, R₂ positioned at the opposite portion with respect to the commontangential line are connected with the smoothing line with respect tothe common tangential line as an axis based on A₁,A₂. At this time, thevalues A₁,A₂ have the same values or different values. Generally, thevalues A₁,A₂ have the same values. In the S type clothoid design, it ismost difficult to determine the values A₁,A₂. When the values A₁,A₂installed at the common tangential line are determined, it is very easyto install the S type clothoid. In particular, it is most important toknow the coordinate of the point 0 that is the intermediate points ofthe center points M1 and M2 of two circles. Namely, two clothoid pointsare set same for thereby determining the positions of the same, so thatthe parameter values A are determined in the S type clothoid. As shownin FIG. 5, in the S type clothoid, the formulas for calculating thevalues A₁,A₂ and the inducing process of the formula will be described.

2. Basic Concept of Calculations of A₁,A₂

1) Necessary specification

The following four values are needed for calculating the parametervalues A₁,A₂ of the S type clothoid using the radius R₁, R₂ of twocircles.

(1) Radius R₁ of the first circle

(2) Radius R₂ of the second circle

(3) Distance D between circumferential portions of two circles on thestraight line connecting the center points of two circles

(4) Value K of the ratio of unknown values A₁,A₂:$K = \frac{A_{2}}{A_{1}}$

At this time, in the values R₁, R₂, it is not needed that the value R₁should be larger, and the value R₂ could be larger than the value R₁.The value A₁ should be the value A₂ with respect to R₁.

Here, the subscripts 1 and 2 represent the values with respect to thevalues R₁, R₂, not the larger or smaller value. Namely, it does notmatter with the size of the radius. The actual values of A₁,A₂ arecalculated based on the value A. Namely, when the value K=1, A₁=A₂, andK is not 1, A₁ is not A₂. When K=0, A₁ is larger than 0, and the valueA₂ is 0.

2) Approaching Method of Formula Induction

The calculation formula of A₁,A₂ is induced from the different fixedrate calculation formula in the clothoid formula.

Namely, since ΔR=Y+R Cos τ−R, Y+Cos τ−R−ΔR=0.

Therefore, the different fixed rate formula for each R₁, R₂ is asfollow.Y ₁ +R ₁Cos τ₁ −R ₁=0  formula 1Y ₂ +R ₂Cos τ₂ −R ₂ −τR ₂=0  formula 2

The formulas 1 and 2 are added, and a result of the same is assumed as afunction F.F=Y ₁ +Y ₂ +R ₁ Cos τ₁ +R ₂Cod τ₂−(R ₁ +ΔR ₁ +R ₂ +ΔR ₂)  formula 3

The different fixed rate formula of ΔR=Y+R Cos τ−R may be expressed as aradius, a smoothing curve length (R, L) or a radius, tangential angle(R, τ), so that the function can be expressed as R₁, R₂, L₁, L₂ and R₁,R₂,τ₁,τ₂. However, the value R₁, R₂ is the given values, namely, theconstant values, so that the function F can be expressed as a functionof L₁,L₂ or τ₁,τ₂.

In the relationship of L₁, L₂ or τ₁,τ₂, in the clothoid formula, A²=RL,and in${K = \frac{A_{2}}{A_{1}}},{K = {\frac{\sqrt{R_{2}\; L_{2}}}{\sqrt{R_{1}\; L_{1}}}.}}$

Since L₂ can be expressed in the formula of L₁, the value of L₂ is thefunction with respect to L₁.

In the same manner, in the clothoid formula, A=√2τR in the${K = \frac{A_{2}}{A_{1}}},{K = \frac{\sqrt{2\;\tau_{2}}\; R_{2}}{\sqrt{2\;\tau_{1}}\; R_{1}}},$so that$\tau_{2} = {\left( {K\;\frac{R_{1}}{R_{2}}} \right)^{2}\;{\tau_{1}.}}$

Assuming that $\begin{matrix}{{t = \left( {K\;\frac{R_{1}}{R_{2}}} \right)^{2}},} & {{formula}\mspace{14mu} 4}\end{matrix}$the formula (5) of τ₂=tτ₁ is obtained.

τ₂ can be expressed in the formula of τ₁, so that the value of τ₂ is thefunction with respect to τ₁.

Therefore, the function F of the formula 3 can be expressed in L₁, or τ₁and in the function of F(L₁) or F(τ₁). In the present invention, thefunction F of the formula 3 will be expressed in the formula of F(τ₁).Therefore, the formula 3 can be expressed in the function of thetangential angle of τ₁ with respect to the radius of R₁.F(τ₁)=Y ₁ +Y ₂ +R ₁Cos τ₁ +R ₂Cos τ₂−(R ₁ +ΔR ₁ +R ₂ +ΔR ₂)  formula 6

It is impossible to obtain the solution of the function F(τ₁) with onlythe formula 6. However, it can be obtained using the non-linealequation. Therefore, the function F(τ₁) is differentiated with respectto r₁, namely, the function F′(τ₁) is obtained based on${{F^{\prime}\;\left( \tau_{1} \right)} = {\frac{\mathbb{d}\;}{\mathbb{d}\tau_{1}}\; F\;\left( \tau_{1} \right)}},$and when dissolving the functions of F′(τ₁) and F′(τ₁)with thenon-lineal equation, it is possible to obtain the value τ₁. When thevalue of τ₁ is obtained, in the clothoid formula, the value A1 can beeasily obtained, and the value A2 can be obtained based on$K = {\frac{A_{2}}{A_{1}}.}$

3) Reference Matters

The function F in the formula 6 is expressed in the function of τ₁. Itdoes not need to calculate in the function of τ₁. When it is calculatedwith the functions of F(L₁) and F′(L₁) with respect to L₁, the sameresult is obtained.

In addition, when the values of L₂ or τ₂ is obtained using F(L₂),F′(L₂)or F(τ₂), F(τ₂), and then the value of A₁, A₂ is obtained, the sameresult is obtained.

When obtaining the resolution of the non-linear equation, it is possibleto fast calculate the value τ₁ using the F(τ₁) as compared tocalculating the τ₁, using F(L₁) when comparing the tangential angle τ₁and the smoothing curve length L₁ In addition, the values of A₁, A₂ areobtained with τ₁ and L₁, and the element values of the clothoid withrespect to two circles are calculated using the values of A₁, A₂, andthe value of D is calculated using the coordinates of the center pointsof two circles. In order to obtain the same value as the value D that isan input specification, the accuracy when the resolution is calculatedbased on the non-linear equation should be smaller as compared to thefunction F(τ₁) in the case of the function F(L₁).

Since L=2τR in the clothoid equation, the above operation should beperformed for the reasons that the value should be significantly smallerwhen the function L is used as compared to when the function τ is usedfor the same result.

Therefore, in the present invention, since the above reasons and theexpression formula of the function are relatively simple, the formula 3is not used in the equation of F(L₁), and the function F(τ₁) is used.

3. Development of the Function of F(τ₁)

The items located in the right sides in the equation ofF(τ₁)=Y₁+Y₂+R₁Cos τ₁,+R₂Cos τ₂−(R₁+ΔR₁+R₂+ΔR₂) may be expressed in theconstant values of R₁, R₂, D, K and the unknown value of τ₁. Namely, allitems located in the right side can be expressed in the function of τ₁.

In the clothoid formula, since A²=2τR², A=√{square root over (2τ)} R andA√{square root over (2)}τ=2τR, and since${X = {{A\;\sqrt{2\;\tau}\;\left( {1 - \frac{\tau^{2}}{10} + \frac{\tau^{4}}{216} - \frac{\tau^{6}}{9360}} \right)} = {2\;\tau\; R\;\left( {1 - \frac{\tau^{2}}{10} + \frac{\tau^{4}}{216} - \frac{\tau^{6}}{9360}} \right)}}},$$X = {2R\;{\left( {1 - \frac{\tau^{3}}{10} + \frac{\tau^{5}}{216} - \frac{\tau^{7}}{9360}} \right).}}$

In the equation of${Y = {\frac{\sqrt{2}}{3}\; A\;\sqrt{\tau^{3}}\;\left( {1 - \frac{\tau^{2}}{14} + \frac{\tau^{4}}{440} - \frac{\tau^{6}}{25200}} \right)}},$${Y = {\frac{\sqrt{2}}{3}\;\sqrt{2\;\tau}\sqrt{\tau^{3}}R\;\left( {1 - \frac{\tau^{2}}{14} + \frac{\tau^{4}}{440} - \frac{\tau^{6}}{25200}} \right)}},{Y = {\frac{2}{3}\; R\;{\left( {\tau^{2} - \frac{\tau^{4}}{14} + \frac{\tau^{6}}{440} - \frac{\tau^{8}}{25200}} \right).}}}$and

In the above formula 6, the first item Y₁ can be expressed in thefunction of τ₁, in the clothoid equation. $\begin{matrix}{Y_{1} = {\frac{2}{3}\; R_{1}\;\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}} & {{Formula}\mspace{14mu} 6a}\end{matrix}$

The second item Y₂ can be expressed in the function of τ₁ in theclothoid equation.

In the formula 5, since r₂=tτ₁, $\begin{matrix}{Y_{2} = {\frac{2}{3}\; R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}}} & {{Formula}\mspace{14mu} 6b}\end{matrix}$

The fourth item R₂ Cos τ₂ can be expressed in the function of τ₁ becauseτ₂=tτ₁ in the formula 5.R ₂Cos τ₂ =R ₂Cos(tτ ₁)  Formula 6c

The last fifth item of (R₁+ΔR₁+R₂+ΔR₂) can be expressed as follows.

In the clothoid equation, since X_(M)=X−R Sin τ, the equations ofX_(M1)=X₁−R Sin τ₁ and X_(M2)=X₂−R₂ Sin τ₂ are obtained.

At this time $\begin{matrix}{{X_{1} = {2R_{1}\;\left( {\tau_{1} - \frac{\tau_{1}^{3}}{10} + \frac{\tau_{1}\; 5}{216} - \frac{\tau_{1}^{7}}{9360}} \right)}},} & {{Formula}\mspace{14mu} 6d}\end{matrix}$and $\begin{matrix}{{X_{2} = {2R_{2}\;\left( {\tau_{2} - \frac{\tau_{2}^{3}}{10} + \frac{\tau_{2}\; 5}{216} - \frac{\tau_{2}^{7}}{9360}} \right)}},} & \;\end{matrix}$and as shown in Formula since τ₂=t τ₁, $\begin{matrix}{X_{2} = {2R_{2}{\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}.}}} & {{Formula}\mspace{14mu} 6e}\end{matrix}$

Assuming that X_(M)=X_(M1)+X_(M2), the equations are X_(M)=X₁−R₁Sinτ₁+X₂−R₂ Sin τ₂ and r₂=t τ₁. Therefore, X_(M) is unknown number, and τ₁,is a function. The above can be expressed as follows.X _(M) =X ₁ −R ₁ Sin τ₁ +X ₂ −R ₂ Sin (tτ ₁)  Formula 6f

In addition, assuming that Y_(M1)=R₁+ΔR₁, Y_(M2)=R₂+ΔR₂,Y_(M)=Y_(M1)+Y_(M2), since the distance between the center points of twocircles is (R₁+D+R₂)²=X_(M) ²+Y_(M) ², andY _(M)=√{square root over ((R ₁ +D+R ₂)² −X _(M) ²)}=Y _(M1) +Y _(M2) =R₁ +ΔR ₁ +R ₂+Δ₂.∴R ₁ +ΔR ₂ +ΔR ₂=√{square root over ((R ₁ +D+R ₂) ² −X _(M) ²)}  Formula6g

In the right side items of FIG. 6, when the formulas 6a, 6b, 6c and 6gare inputted into the first, second, fourth and fifth items, as shown inthe formula 6, the unknown number remains τ₁(Formula 7). The lastformulas of the function of F(τ₁) is as follows. $\begin{matrix}\begin{matrix}{{F\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{1}\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}} +}} \\{{\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}} +} \\{{R_{1}{Cos}\mspace{11mu}\tau_{1}} + {R_{2}{Cos}\mspace{11mu}\left( {t\;\tau_{1}} \right)} - {\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}.}}\end{matrix} & {{formula}\mspace{14mu} 7}\end{matrix}$

-   -   wherein in the formula 7, X_(M) is X_(M)=X₁−R₁ Sin        τ₁+X₂−R₂Sin(tτ₁) based on Formula 6a, and X₁,X₂ can be obtained        based on formulas 6d and 6e.

Therefore, the function F(τ₁) may use the formula 7 or the formula 6. Inthe case that the formula 7 is used, the formula is physically too long.In addition, in the case that the formula 6 is used, the values of theformulas 6a, 6b, 6c and 6e may be inputted.

Here, when computing the function, in √{square root over((R₁+D+R₂)²−X_(M) ²)}≦0, the tangential angle value τ₁ should beadjusted to a proper value, the equations of$\left. {\tau_{1} = \frac{\tau_{1}}{2}} \right)$F(τ₁)

F′(τ₁) should be calculated.

4. Development of the Function F′(τ₁)

The function F(τ₁) is differentiated with respect to the value τ₁, forthereby forming F′(τ₁), so that${F^{\prime}\left( \tau_{1} \right)} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{F\left( \tau_{1} \right)}}$is obtained.

At this time, the function F(τ₁)

should be differentiated using the formula 7, not the formula 6.$\begin{matrix}\begin{matrix}{{\therefore{F^{\prime}\left( \tau_{1} \right)}} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {Y_{1} + Y_{2} + {R_{1}{Cos}\mspace{11mu}\tau_{1}} + {R_{2}{Cos}\mspace{11mu}\tau_{2}} -} \right.}} \\\left. \sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}} \right\}\end{matrix} & {{Formula}\mspace{14mu} 8}\end{matrix}$

Each item at the right side of the formula 8 will be differentiated withrespect to the value τ₁.

When differentiating the first item Y₁ with respect to r₁,

In the formula 6a, $\begin{matrix}{{{Y_{1} = {\frac{2}{3}{R_{1}\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}}},{and}}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( Y_{1} \right)} = {\frac{2}{3}{{R_{1}\left( {{2\tau_{1}} - \frac{4\;\tau_{1}^{3}}{14} + \frac{6\tau_{1}^{5}}{440} - \frac{8\tau_{1}^{7}}{25200}} \right)}.}}}} & {{Formula}\mspace{14mu} 8a}\end{matrix}$

When differentiating the second item Y₂ with respect to τ₁, In theformula 6b),${Y_{2} = {\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)8}{25200}} \right\}}},$and $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( Y_{2} \right)} = {\frac{2}{3}{{R_{2}\left( {{2t^{2}\tau_{1}} - \frac{4t^{4}\tau_{1}^{3}}{14} + \frac{6t^{6}\tau_{1}^{5}}{440} - \frac{8t^{8}\tau_{1}^{7}}{25200}} \right)}.}}} & {{formula}\mspace{14mu} 8b}\end{matrix}$

When differentiating the third item R₂Cos τ₂, with respect to τ₁,$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{1}{Cos}\mspace{11mu}\tau_{1}} \right)} = {{- R_{1}}{Sin}\mspace{11mu}{\tau_{1}.}}} & {{formula}\mspace{14mu} 8c}\end{matrix}$

When differentiating the fourth item R₂ Cos τ₂ with respect to τ₁, wecan get τ₂=tτ₁ based on the formula 5, and $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{2}{Cos}\mspace{11mu}\tau_{2}} \right)} = {{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {R_{2}{Cos}\;\left( {t\;\tau_{1}} \right)} \right\}} = {{- R_{2}}t\;{{{Sin}\left( {t\;\tau_{1}} \right)}.}}}} & {{formula}\mspace{14mu} 8d}\end{matrix}$

Now, when differentiating the fifth item √{square root over((R₁+D+R₂)²−X_(M) ²)} with respect to τ₁, $\begin{matrix}{{{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\;\left\{ {\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}} \right\}^{\overset{1}{\Leftrightarrow}2}}},{and}}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}} = {{\frac{1}{2}\left\{ {\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}} \right\}^{\overset{- 1}{\Leftrightarrow}2}*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {- X_{M}^{2}} \right)} = {{{\frac{1}{2\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*\left( {{- 2}X_{M}} \right)*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}}\therefore{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}} = {\frac{- X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{X_{M}.}}}}}} & {{formula}\mspace{14mu} 8e}\end{matrix}$

In the formula 6f, the value X_(M) is the function of the unknown valueτ₁,$\frac{- X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}$may be substituted with the constant values of R₁, R₂, D, K and τ₁, itis the function of τ₁. Therefore, the equation of$\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}$may be developed with the function of τ₁.

In the formula 6f, since it is X_(M)=X₁−R₁Sin τ₁+X₂−R₂ Sin (tτ₁),$\begin{matrix}{{{we}\mspace{14mu}{can}\mspace{14mu}{get}\mspace{14mu}\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{\left\{ {X_{1} - {R_{1}{Sin}\mspace{11mu}\tau_{1}} + X_{2} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}}} \right\}.}}} & {{formula}\mspace{14mu} 8f}\end{matrix}$

In the formula 6d, since it is${X_{1} = {2{R_{1}\left( {\tau_{1} - \frac{\tau_{1}^{3}}{10} + \frac{\tau_{1}^{5}}{216} - \frac{\tau_{1}^{7}}{9360}} \right)}}},$we can get $\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{1}} = {2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}}},} & {{formula}\mspace{14mu} 8f\text{-}1}\end{matrix}$and $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{1}{Sin}\;\tau_{1}} \right)} = {R_{1}{Cos}\;{\tau_{1}.}}} & {{formula}\mspace{14mu} 8f\text{-}2}\end{matrix}$

In the formula 6e, it is${X_{2} = {2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}}},$$\begin{matrix}{{we}\mspace{14mu}{can}\mspace{14mu}{get}} & \; \\{{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{2}} = {2R_{2}t\left\{ {1 - \frac{3\left( {t\;\tau_{1}} \right)^{2}}{10} + \frac{5\left( {t\;\tau} \right)^{4}}{216} - \frac{7\left( {t\;\tau_{1}} \right)^{6}}{9360}} \right\}}},} & {{formula}\mspace{14mu} 8f\text{-}3} \\{and} & \;\end{matrix}$ $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} \right\}} = {R_{2}t\mspace{14mu}{{{Cos}\left( {t\;\tau_{1}} \right)}.}}} & {{formula}\mspace{14mu} 8f\text{-}4}\end{matrix}$

When inputting the formulas 8f-1 through 8f-4 into the formula 8f, wecan get $\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}$as follows. $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}} = {{2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + {2R_{2}t\left\{ {1 - \frac{3\left( {t\;\tau_{1}} \right)^{2}}{10} + \frac{5\left( {t\;\tau_{1}} \right)^{4}}{216} - \frac{7\left( {t\;\tau_{1}} \right)^{6}}{9360}} \right\}} - {R_{2}t{\left\{ {{Cos}\left( {t\;\tau_{1}} \right)} \right\}.}}}} & {{formula}\mspace{14mu} 8g}\end{matrix}$

Therefore, when inputting $\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}$of the formula 8g into the formula 8e, we can get the fifth item of theformula 8 of $\begin{matrix}{{\frac{\mathbb{d}\;}{\mathbb{d}\tau_{1}}\;\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}} = {\frac{- X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*{\left\{ {{2R_{1}\;\left( {1 - \frac{3\;\tau_{1}^{2}}{10} + \frac{5\;\tau_{1}^{4}}{216} - \frac{7\;\tau_{1}^{6}}{9360}} \right)} - {R_{1}\;{Cos}\mspace{11mu}\tau_{1}} + {2R_{2}\; t\left\{ {1 - \frac{3\;\left( {t\;\tau_{1}} \right)^{2}}{10} + \frac{5\;\left( {t\;\tau_{1}} \right)^{4}}{216} - \frac{7\;\left( {t\;\tau_{1}} \right)^{6}}{9360}} \right\}} - {R_{2}\; t\mspace{11mu}{Cos}\;\left( {t\;\tau_{1}} \right)}} \right\}.}}} & {{formula}\mspace{14mu} 8h}\end{matrix}$

When inputting the differentiated formulas into the formula 8, we canget the final equation of${F^{\prime}\left( \tau_{1} \right)} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{F\left( \tau_{1} \right)}}$as follows. $\begin{matrix}{{F^{\prime}\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{1}\left( {{2\tau_{1}} - \frac{4\tau_{1}^{3}}{14} + \frac{6\tau_{1}^{5}}{440} - \frac{8\tau_{1}^{7}}{25200}} \right)}} + {\frac{2}{3}{R_{2}\left( {{2t^{2}\tau_{1}} - \frac{4t^{4}\tau_{1}^{3}}{14} + \frac{6t^{6}\tau_{1}^{5}}{440} - \frac{8t^{8}\tau_{1}^{7}}{25200}} \right)}} - {R_{1}\mspace{11mu}{Sin}\;\tau_{1}} - {R_{2}t\mspace{14mu}{{Sin}\left( {t\;\tau_{1}} \right)}} + {\frac{X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*{\left\{ {{2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + {2R_{2}t\left\{ {1 - \frac{3\left( {t\;\tau_{1}} \right)^{2}}{10} + \frac{5\left( {t\;\tau_{1}} \right)^{4}}{216} - \frac{7\left( {t\;\tau_{1}} \right)^{6}}{9360}} \right\}} - {R_{2}t\mspace{14mu}{{Cos}\left( {t\;\tau_{1}} \right)}}} \right\}.}}}} & {{formula}\mspace{14mu} 9}\end{matrix}$

-   -   here, in the formula 9, the value of X_(M) can be calculated        with the formula 6f and can be inputted.

5. Computation Method of A₁, A₂ Using Non-Linear Equation

When resolving the formula of F(τ₁), F′(τ₁) in the formulas 7 and 9based on Newton-Raphson), we can get the tangential angle of τ₁ withrespect to R₁, we can easily get A₁,A₂ with respect to R₁, R₂. Now, theflow chart of the method for getting the resolution based onNewton-Raphson is shown in FIG. 7. We assume that the accuracy of thecalculation of the tangential angle τ₁, is 10⁻⁶. The functions F and F′can be expressed with the smoothing curve length L₁, not with thetangential angle τ₁ as follows.${F\left( L_{1} \right)} = {{L_{1}^{2}\left( {1 - \frac{L_{1}^{2}}{56R_{1}^{2}} + \frac{L_{1}^{4}}{7040R_{1}^{4}} - \frac{L_{1}^{6}}{1612800R_{1}^{6}}} \right)} + {\frac{L_{2}^{2}}{6R_{2}}\left( {1 - \frac{L_{2}^{2}}{56R_{2}^{2}} + \frac{L_{2}^{4}}{7040R_{2}^{4}} - \frac{L_{2}^{6}}{1612800R_{2}^{6}}} \right)} + {R_{1}{{Cos}\left( \frac{L_{1}}{2R_{1}} \right)}} + {R_{2}{{Cos}\left( {t\;\frac{L_{1}}{2R_{2}}} \right)}} - \sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}$${F^{\prime}\left( L_{1} \right)} = {{\frac{1}{6R_{1}}\left( {{2L_{1}} - \frac{4L_{1}^{3}}{56R_{1}^{2}} + \frac{6L_{1}^{5}}{7040R_{1}^{4}} - \frac{8L_{1}^{7}}{161280R_{1}^{6}}} \right)} + {\frac{1}{6R_{2}}\left( {{2t^{2}L_{1}} - \frac{4t^{4}L_{1}^{3}}{56R_{2}^{2}} + \frac{6t^{6}L_{1}^{5}}{7040R_{2}^{4}} - \frac{8t^{8}L_{1}^{7}}{161280R_{2}^{6}}} \right)} - {\frac{1}{2}{{Sin}\left( \frac{L_{1}}{2R_{1}} \right)}} - {t\;{{Sin}\left( {t\;\frac{L_{1}}{2R_{2}}} \right)}} +}$

Here, in the case that it is used in the function of L₁, in order toobtain the same value A₁ as the case of τ₁, we can get$\tau = \frac{L}{2R}$based on the clothoid formula, so that in the above flow chart, theaccuracy degree should be $10^{- 6}*\frac{1}{2R_{1}}$instead of 10⁻⁶.

The flow chart of the S type clothoid parameter calculation method ofFIG. 7 will be described.

In a method for calculating a S shape clothoid parameter including aunknown clothoid parameter A adapted to radius (R₁ R₂ of two circles,the shortest distance between circumferential portions of two circlesand two circles, there is a method for calculating a S shape clothoidparameter, comprising a step in which an initial value of a tangentialangle τ₁ is set; a step in which the value of (R₁+D+R ₂)²−Xm² iscalculated in such a manner that the tangential angle (τ₁) is compared,and when a result of the comparison is below 0°, since it means there isnot any resolution, the process is stopped, and when a result of thesame is over 0°, the process is continued; a step in which the value of(R₁+D+R₂)²−Xm² is compared with 0, and when a result of the comparisonis below 0, the tangential angle is properly adjusted, and the routinegoes back to the step for setting the initial value of the tangentialvalue of (τ₁), and when a result of the same is over 0, a differentfixed rate formula is set up with respect to two circles, and oneformula is formed by adding the left and right items in two formulas,for thereby obtaining a tangential angle (τ₁); a step in which thefunction F(τ₁) of the tangential angle (τ₁) and the function aredifferentiated with the tangential angle (τ₁)for thereby calculating thedifferential function F′(τ₁); a step in which the ratio[G=(F(τ₁)/F′(τ₁)] of two functions of [(F(τ₁), F′(τ₁)] are calculated;and a step in which the absolute value of the ratio(G) is compared witha permissible error (10-6), and as a result of the comparison when it isover the permissible error, the initial value of the tangential value(τ_(s)) is set, and the routine is fed back to the next step of the stepthat the initial value is set, and when it is below the permissibleerror, the tangential angle (τ₁) is determined, and the parameter valueA is calculated using the tangential angle (τ₁).

6. Calculation Examples of A₁, A₂ in S type Clothoid 1) in the case thatA₁=A₂(namely, Symmetrical Type).

EXAMPLE 1

When input specification is R₁=250, R₂=200, D≈15.088, K=1,

As a result of the calculation, A₁=A₂=149.99802855 can be calculatedwith about four calculations.

The distance between the center points of two circles using the elementvalues of the clothoid of two circles can be calculated using A₁, A₂ forthereby obtaining R₁+D+R₂.

In the case that we use L₁ instead of τ₁, the value having a desiredaccuracy can be obtained with about 40˜50 calculations. The distancebetween the center points of two circles can be calculated like the caseof τ₁ using A₁, A₂ and the clothoid element values of two circles, sothat there is a very small difference of about 10⁻⁸˜10⁻¹⁰ as comparedwith R₁+D+R₂.

EXAMPLE 2 In the case that R₁=200, R₂=120, D=105.6,

As a result of the above algorithm, we can get A₁≈A₂≈190.00785.

2) In the case of A₁≠A₂(Namely, Non-Symmetrical)

Input data: in the case of${R_{1} = 230},\;{R_{2} = 180},{D = 26.5},{K = {\frac{A_{2}}{A_{1}} = 0.8}},$

A result of the calculation: five calculationsA1=181.340639936522, A2=145.072511949218

-   -   verification is        $K = {\frac{A_{2}}{A_{1}} = {\frac{145.072511949218}{181.340639936522} = 0.8}}$        Xm1=71.2582950417867, Xm2=58.2562467892329, XM=129.51454183102        Ym1=233.690522740909, Ym2=183.152653350893, YM=416.843176091802

When calculating the value of D, we can get √{square root over (X_(M)²+Y_(M) ²)}=436.5 in (R₁+D+R₂)²=X_(M) ²+Y_(M) ², andD=436.5−(230+180)=26.5. The above value is matched with the values givenas the specification.

Second Embodiment Calculation Method of Egg Type Clothoid Parameter inEgg Type Design

1. Shape of Egg Type

When the egg type is installed, we can get the final shape of FIG. 6. Atthis time, in the clothoid curve, the range of the radius of R_(l)˜R_(s)among the smoothing curve generated by the parameter A_(E) can be used,and the larger circle and the smaller circle are connected using thesmoothing curve.

2. Basic Concept of the Calculation of the Parameter A in the Egg Type

In order to geometrically explain the calculation formulas in the eggtype, the symbols related to the egg type clothoid curve in FIG. 6 areindicated as the subscripts of _(E1), _(E2), and the data with respectto the larger circle in the egg type clothoid curve are indicated withthe subscripts of ₁, and the data of the smaller circle are indicatedwith the subscripts of ₂.

From now on, the subscript of ₁ represents the larger circle, and thesubscript of ₂ represents the smaller circle. In the egg type, theclassifications of the larger and smaller circles are very important. Asthe conditions of the egg type, there are {circle around (1)} the largercircle should fully contain the smaller circle. {circle around (2)} thecenters of two circles should not be the same (non-concentrical). Asshown in FIG. 6, the key points in the egg type design are to recognizethe coordinates of KAE and KEE in which the egg type clothoid contactswith the larger circle and the smaller circle. This coordinate can beeasily obtained when knowing the egg type clothoid parameter value A.

1) Specification Needed

The following three values are needed in order to calculate the egg typeclothoid parameter A using the radius of R₁,R₂ of two circles.

-   -   {circle around (1)} Radius R₁ of larger circle    -   {circle around (2)} Radius of R₂ of smaller circle    -   {circle around (3)} Minimum distance between larger circle and        smaller circle

2) Approaching Method of Formula Induction

The calculation formula of the value A is induced from the differentfixed rate calculation formula in the clothoid.

Namely, since ΔR=Y+R Cos τ−R, we can get Y+R Cos τ−R−τR=0. Therefore,when adapting the unknown parameter A, the different fixed rate formulawith respect to R₁,R₂ can be expressed as follows.Y ₁ +R ₁ Cos τ₁ −R ₁ τR=  formula 1Y ₂ +R ₂ Cos τ₂ −R ₂ −τR ₂₌₀—formula 2

In the egg type, since the different fixed rate of the smaller circlewith respect to the parameter A is always larger than the differentfixed rate with respect to the larger circle, the formula 1 issubtracted from the formula 2, and assuming that the result of thecalculation is the function F, we can get the following.F=Y ₂ −Y ₁ +R ₂ Cos τ₂ −R ₁ Cos τ₁+(R ₁ +ΔR ₁)−(R ₂ +ΔR ₂)=0  formula 3

The different fixed rate formula of ΔR=Y+R Cos τ−R can be expressed withradius, smoothing curve length(R, L) or radius and tangential angle(R,τ), so that the function F can be expressed with R₁, R₂,L₁, L₂ or R₁,R₂,τ₁,τ₂. Since the values of R₁, R₂ are given values, namely, constantvalues, the function F can be expressed in the function of L₁, L₂ orτ₁,τ₂.

In the relationship between the values of L₁ and L₂, since R₁L₁=R₂L₂ inthe clothoid formula of A²=RL, we can get$L_{2} = {\frac{R_{1}}{R_{2}}{L_{1}.}}$Namely, since the value L₂ can be expressed in the formula of L₁, thefunction L₂ is the function with respect to L₁. In the same manner,since 2τ₁R₁ ²=2τ₂R₂ ² in the clothoid formula of A²=2τR², 2τ₁R₁ ²=2τ₂R₂², we can get$\tau_{2} = {\left( \frac{R_{1}}{R_{2}} \right)^{2}{\tau_{1}.}}$

-   -   assuming that $\begin{matrix}        {{t = \left( \frac{R_{1}}{R_{2}} \right)^{2}},} & {{formula}\mspace{14mu} 4}        \end{matrix}$        τ₂ =tτ ₁  formula 5.

Therefore, τ₂ can be expressed in the formula of τ₁, and the function τ₂is the function with respect to τ₁.

Therefore, since the function F of the formula 3 can be expressed withL₁ or τ₁, it can be expressed in the functions of F(L₁) or F(τ₁).

In the present invention, the function F of formula 3 can be developedin the formula of F(τ₁). Therefore, the formula 3 can be expressed inthe function of the tangential angle τ₁ with respect to the radius R₁.F(τ₁)=Y ₂ −Y ₁ +R ₂ Cos τ₂ −R ₁Cos τ₁+{(R ₁ +ΔR ₁)−(R ₂ +ΔR ₂)}  formula6

It is impossible to obtain the resolution of the function F(τ₁) withonly the formula 6. However, the resolution can be obtained using thenon-linear equation. Namely, when differentiating the function F(τ₁)with respect to τ₁, namely,${{F^{\prime}\left( \tau_{1} \right)} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{F\left( \tau_{1} \right)}}},$for thereby forming the function of F′(τ₁). When resolving the functionsF(τ₁) and F′(τ₁) based on the non-linear equation, we can obtain thevalue of τ₁. When the value of τ₁ is obtained, we can easily obtain thevalue A based on the clothoid formula.

3) Reference Matters

In the formula 6, we obtained the value of τ₁ by expressing the functionin the function of τ₁. It does not need to always calculate the same inthe function of τ₁ Namely, even when it is resolved in the functions ofF(L₁, F′(L₁) with respect to the function of L₁, we can get the sameresult. In addition, the values of L₂ or τ₂ can be obtained using thefunctions of F(L₂), F′(L₂) or F(τ₂), F(τ₂), and then the value A isobtained. In this case, we can also get the same result.

When obtaining the resolution of the non-linear equation and comparingthe tangential angle τ₁ and smoothing curve L₁, it is possible to fastercalculate the value of τ₁, using the function F(τ₁) as compared to whencalculating the same using F(L₁). In addition, the value is obtainedusing τ₁, L₁, and then the element values of the clothoid of two circlesare computed using the calculated values A, and the value D iscalculated using the coordinates of the center points of two circles. Inorder to obtain the same result as the value D that is in the inputspecification, the accuracy should be significantly smaller as comparedto the function F(τ₁) in the case of the function F(L₁) when calculatingin the non-linear equation.

In the clothoid formula, since it is L=2τR in the case that the functionof L is used for the same result, it is needed to obtain the valuesignificantly smaller as compared to when using the function τ.

Therefore, since the above reasons and the expressions of the functionsare relatively simple in the data of the present invention, the functionof F(τ₁) is used without using the formula 3 as the function of F(L₁).

3. Development of the Function F(τ₁)

Each items placed in the formula 6 of F(τ₁)=Y₂−Y₁+R₂ Cos τ₂−R₁ Cosτ₁+{(R₁+ΔR₁)−(R₂+ΔR₂)} can be expressed with the constant values of R₁,R₂, D and the unknown value of τ₁ Namely, all items placed in the rightside can be expressed in the function of τ₁.

In the clothoid formula, since it is A²=2τR², we can get A=√{square rootover (2τ)}R therefore, A√{square root over (2τ)}=2τR.$X = {{A\sqrt{2\tau}\left( {1 - \frac{\,\tau^{2}}{10} + \frac{\tau^{4}}{216} - \frac{\,\tau^{6}}{9360}} \right)} = {{{2\tau\;{R\left( {1 - \frac{\tau^{2}}{10} + \frac{\tau^{4}}{216} - \frac{\tau^{6}}{9360}} \right)}}\therefore X} = {2{R\left( {\tau - \frac{\,\tau^{3}}{10} + \frac{\tau^{5}}{216} - \frac{\,\tau^{7}}{9360}} \right)}}}}$$Y = {{\frac{\sqrt{2}}{3}A\sqrt{\tau^{3}}\left( {1 - \frac{\,\tau^{2}}{14} + \frac{\tau^{4}}{440} - \frac{\,\tau^{6}}{25200}} \right)} = {{{\frac{\sqrt{2}}{3}A\sqrt{2\tau}\sqrt{\tau^{3}}{R\left( {1 - \frac{\,\tau^{2}}{14} + \frac{\tau^{4}}{440} - \frac{\,\tau^{6}}{25200}} \right)}}\therefore Y} = {\frac{2}{3}{R\left( {\tau^{2} - \frac{\,\tau^{4}}{14} + \frac{\tau^{6}}{440} - \frac{\,\tau^{8}}{25200}} \right)}}}}$

The first item Y₂ can be expressed in the function of τ₁ in theclothoid.

In the formula 5, since it is τ₂=tτ₁, $\begin{matrix}{Y_{2} = {\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}}} & {{formula}\mspace{14mu} 6a}\end{matrix}$

The second item Y₁ can be expressed in the function of τ₁ in theclothoid. $\begin{matrix}{Y_{1} = {\frac{2}{3}{R_{1}\left( \;{\tau_{1}^{2} - \frac{\;\tau_{1}^{4}}{14} + \frac{\;\tau_{1}^{6}}{440} - \frac{\;\tau_{1}^{8}}{25200}} \right)}}} & {{formula}\mspace{14mu} 6b}\end{matrix}$

The third item R₂ Cos τ₂ can be expressed in the function of τ₁ becauseit is τ₂=tτ₁ in the formula 5.R ₂ Cos τ₂ =R ₂ Cos(tτ ₁)  formula 6c

The last fifth item (R₁+ΔR₁)−(R₂+ΔR₂) can be expressed as follows.

Assuming that the coordinate of the unknown center point of the largercircle is M₁, and the coordinate of the unknown center point of thesmaller circle is M₂, the distance {overscore (M₁M₂)} of the centerpoints of two circles is M₁M₂P

({overscore (M₁M₂)})²=({overscore (M₁P)})²+({overscore (M₂P)})² in thetriangle M₁M₂P of FIG. 6, we can get{overscore (M ¹ M ² )}=( R ₁ −R ₂ −D), {overscore (M ¹ )} P=(R ₁ +ΔR₁)−(R ₂ +ΔR ₂), {overscore (M ² )} P=X _(M2) −X _(M1).

Therefore, (R₁+ΔR₁)−(R₂+ΔR₂)=√{square root over (({overscore(M₁M₂)})})²−(X_(M2)−X_(M1))².∴(R ₁ +ΔR ₁)−(R ₂ +ΔR ₂)=√{square root over ((R ₁ −R ₂ −D)²−(X _(M2) −X_(M1))²)}{square root over ((R ₁ −R ₂ −D)²−(X _(M2) −X_(M1))²)}  formula 6d

In the clothoid formula of X_(M)=X−R₁ Sin τ₁ it is X_(M1)=X₁−R₁Sin τ₁,X_(M2)=X₂−R₂ Sin τ₂. $\begin{matrix}{{Y_{1} = {2{R_{1}\left( \;{\tau_{1} - \frac{\;\tau_{1}^{3}}{10} + \frac{\;\tau_{1}^{5}}{216} - \frac{\;\tau_{1}^{7}}{9360}} \right)}}},} & {{formula}\mspace{14mu} 6d\text{-}1}\end{matrix}$and

In${X_{2} = {2{R_{2}\left( \;{\tau_{2} - \frac{\;\tau_{2}^{3}}{10} + \frac{\;\tau_{2}^{5}}{216} - \frac{\;\tau_{2}^{7}}{9360}} \right)}}},$since it is τ₂=tτ₁ based on the formula 5, it is $\begin{matrix}{X_{2} = {2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}}} & {{formula}\mspace{14mu} 6d\text{-}2}\end{matrix}$

Therefore, when the formulas 6d-1 and 6d-2 are inputted into the formula6d, we can get τ₂=tτ₁, so that the equation of X_(Ml)−X_(M1) is asfollows. $\begin{matrix}{{X_{M2} - X_{M1}} = {{2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} - \left\{ {{2{R_{1}\left( \;{\tau_{1} - \frac{\;\tau_{1}^{3}}{10} + \frac{\;\tau_{1}^{5}}{216} - \frac{\;\tau_{1}^{7}}{9360}} \right)}} - {R_{1}S\; i\; n\;\tau_{1}}} \right\}}} & {{formula}\mspace{14mu} 6d\text{-}3}\end{matrix}$

Therefore, the formula 6d can be expressed as follows.(R ₁ +ΔR ₁)−(R ₂ +ΔR ₂)=√{square root over ((R ₁ −R ₂ −D)²−a²)}  formula6e

-   -   wherein the value of a in the formula 6e is as follows.        $a = {{2R_{2}\left\{ {\left( {t\; r_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} - \left\{ {{2{R_{1}\left( \;{\tau_{1} - \frac{\;\tau_{1}^{3}}{10} + \frac{\;\tau_{1}^{5}}{216} - \frac{\;\tau_{1}^{7}}{9360}} \right)}} - {R_{1}S\; i\; n\;\tau_{1}}} \right\}}$

When the formulas 6a, 6b, 6c and 6e are inputted into the first, second,third and fifth items placed in the right side of the formula 6, we canget the function F(τ₁) as follows. $\begin{matrix}{{F\left( \tau_{1} \right)} = {{\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}} - {\frac{2}{3}{R_{1}\left( \;{\tau_{1}^{2} - \frac{\;\tau_{1}^{4}}{14} + \frac{\;\tau_{1}^{6}}{440} - \frac{\;\tau_{1}^{8}}{25200}} \right)}} + {R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + \sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - a^{2}}}} & {{formula}\mspace{14mu} 7}\end{matrix}$

In the formula 7,$a = {{2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} - \left\{ {{2{R_{1}\left( {\tau_{1} - \frac{\tau_{1}^{3}}{10} + \frac{\tau_{1}^{5}}{216} - \frac{\tau_{1}^{7}}{9360}} \right)}} - {R_{1}{Sin}\;\tau_{1}}} \right\}}$

When using the function F(τ₁), if the formula 7 is too long and is noteasy to use, the values of the formulas 6a, 6b, 6c and 6e arecalculated, and inputted into the formula 6 for thereby achieving aneasier use.

It is important to recognize that when calculating the function F(τ₁),in the case that (R₁+ΔR₁)−(R₂+ΔR ₂)≦0 of the formula 6e, the tangentialangle τ₁, should be adjusted with a proper value (for example:$\left. {\tau_{1} = \frac{\tau_{1}}{2}} \right),$and the functions of F(τ₁) and F′(τ₁) should be calculated again.

4. Development of the Function F′(τ₁)

The function of F(τ₁) is differentiated with respect to τ₁, for therebyforming the function F′(τ₁), so that we can get${F^{\prime}\left( \tau_{1} \right)} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{{F\left( \tau_{1} \right)}.}}$

At this time, since the unknown values of F′(τ₁) and F(τ₁) should beexpressed with only τ₁, when differentiating using the formula 6, eachitem placed in the right side should be independently differentiated,and should be expressed with only the constant numbers of R₁, R₂,D, Kand the unknown number τ₁ or it is needed to differentiate based on theformula 7.

Since differentiating the formula 7 is more easy, the formula 7 isapplied in the following description.

When the formula 7 itself is differentiated, since the formula of theitem of √{square root over ( )} is complicated, it is needed todifferently express the item of √{square root over ( )} as follows.Namely, it is same as the √{square root over((R₁−R₂−²−(X_(M2)−X_(M1))²)} of the formula 6d, the formula 7 can beexpressed as follows. $\begin{matrix}{{F\left( \tau_{1} \right)} = {{\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}} - \left\{ {\frac{2}{3}{R_{1}\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}} \right\} + {R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + {\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - \left( {X_{M2} - X_{M1}} \right)^{2}}.}}} & {{formula}\mspace{11mu} 8}\end{matrix}$

Each item placed in the right side of the formula 8 can bedifferentiated with respect to τ₁.

When the first item is differentiated with respect to τ₁, we can get$\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}} = {\frac{2}{3}{R_{2}\left( {{2t^{2}\tau_{1}} - \frac{4t^{4}\tau_{1}^{3}}{14} + \frac{6t^{6}\tau_{1}^{5}}{440} - \frac{8t^{8}\tau_{1}^{7}}{25200}} \right)}}},{and}} & {{formula}\mspace{11mu} 8a}\end{matrix}$

-   -   when the second item is differentiated with respect to τ₁, we        can get $\begin{matrix}        {{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {\frac{2}{3}{R_{1}\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}} \right\}} = {\frac{2}{3}{{R_{1}\left( {{2\tau_{1}} - \frac{4\tau_{1}^{3}}{14} + \frac{6\tau_{1}^{5}}{440} - \frac{8\tau_{1}^{7}}{25200}} \right)}.}}} & {{formula}\mspace{11mu} 8b}        \end{matrix}$

In addition, the third item is differentiated with respect to τ₁, we canget $\begin{matrix}{\;{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} \right\}} = {{- R_{2}}t\;{{{Sin}\left( {t\;\tau_{1}} \right)}.}}}} & {{formula}\mspace{14mu} 8c}\end{matrix}$

And, when the fourth item is differentiated with respect to r₁, we canget $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{1}{Cos}\;\tau_{1}} \right)} = {{- R_{1}}\;{Sin}\;{\tau_{1}.}}} & {{formula}\mspace{11mu} 8d}\end{matrix}$

In addition, the fifth item of √{square root over((R₁−R₂−D)²−(X_(Ml)−X_(M1))²)}{square root over((R₁−R₂−D)²−(X_(Ml)−X_(M1))²)} is differentiated with respect to τ₁,assuming

-   -   X_(M)=X_(M2)−X_(M1), since $\begin{matrix}        {{{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}} \right\}^{\overset{1\mspace{11mu}}{\Longleftrightarrow 2}}}},{{we}\mspace{14mu}{can}\mspace{14mu}{get}}}{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\{ {\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}} \right\}^{\frac{1}{2}}} = {{\frac{1}{2}\left\{ {\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}} \right\}^{\overset{{- 1}\mspace{11mu}}{\Longleftrightarrow 2}}*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {- X_{M}^{2}} \right)} = {{{\frac{1}{2\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}}*\left( {{- 2}X_{M}} \right)*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}}\therefore{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}}} = {\frac{- X_{M}}{\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}}*\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}}}}} & {{formula}\mspace{14mu} 8e}        \end{matrix}$

Now $\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}$will be developed with the function of τ₁ since it isX_(M)=X_(Ml)−X_(M1) and X_(Ml)−X_(M1) is the formula 6d-3, we can get$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}} = {{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left\lbrack {{2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right\}} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} - \left\{ {{2{R_{1}\left( {\tau_{1} - \frac{\tau_{1}^{3}}{10} + \frac{\tau_{1}^{5}}{216} - \frac{\tau_{1}^{7}}{9360}} \right)}} - {R_{1}{Sin}\;\tau_{1}}} \right\}} \right\rbrack}\therefore{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}X_{M}}} = {{2R_{2}\left\{ {t - \frac{3t^{3}\tau_{1}^{2}}{10} + \frac{5t^{5}\tau_{1}^{4}}{216} - \frac{7t^{7}\tau_{1}^{6}}{9360}} \right\}} - {{tR}_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - {2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} + {R_{1}{Cos}\;\tau_{1}}}}} & {{formula}\mspace{14mu} 8f}\end{matrix}$

Now when the formula 8f is inputted into the formula 8e, the followingformula 9 is obtained. $\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}} = {\frac{- X_{M}}{\sqrt{\left( {R_{1} - R_{2} - D} \right)^{2} - X_{M}^{2}}}*\mspace{79mu}\left\lbrack {2{\quad{R_{2}{\quad\left. \quad{\left\{ {t - \frac{3t^{3}\tau_{1}^{2}}{10} + \frac{5t^{5}\tau_{1}^{4}}{216} - \frac{7t^{7}\tau_{1}^{6}}{9360}} \right\} - {t\; R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - {2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} + {R_{1}{Cos}\;\tau_{1}}} \right\rbrack}}}} \right.}} & {{formula}\mspace{14mu} 9}\end{matrix}$

Therefore, the last formula of${F^{\prime}\left( \tau_{1} \right)} = {\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{F\left( \tau_{1} \right)}}$is as follows. $\begin{matrix}\begin{matrix}{{F^{\prime}\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{2}\left( {{2t^{2}\tau_{1}} - \frac{4t^{4}\tau_{1}^{3}}{14} + \frac{6t^{6}\tau_{1}^{5}}{440} - \frac{8t^{8}\tau_{1}^{7}}{25200}} \right)}} -}} \\{{\frac{2}{3}{R_{1}\left( {{2\tau_{1}} - \frac{4\tau_{1}^{3}}{14} + \frac{6\tau_{1}^{5}}{440} - \frac{8\tau_{1}^{7}}{25200}} \right)}} - {R_{2}t\;{{Sin}\left( {t\;\tau_{1}} \right)}} +} \\{{R_{1}{Sin}\;\tau_{1}} - {\frac{X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*}} \\{\left\lbrack {{2R_{2}\left\{ {t - \frac{3t^{3}\tau_{1}^{2}}{10} + \frac{5t^{5}\tau_{1}^{4}}{216} - \frac{7t^{7}\tau_{1}^{6}}{9360}} \right\}} - {t\; R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} -} \right.} \\{\left. {{2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} + {R_{1}{Cos}\;\tau_{1}}} \right\rbrack\;}\end{matrix} & {{formula}\mspace{11mu} 10}\end{matrix}$

-   -   wherein, in the formula 10, it is X_(M)=X_(Ml)−X_(M1), and the        formula of X_(Ml)−X_(M1) is the formula 6d-3.

The formula 10 is obtained by developing the function of F′(τ₁). When itis too long to use it, the values of each item in the formula 6 or 7 areindependently calculated for thereby calculating F′(τ₁). Whencalculating the same by differentiating the formulas 10 and 6, it isneeded to consider to omit ± in each item in the formula 10, we can getthe first item of${\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( Y_{2} \right)},$the second item of${\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( Y_{1} \right)},$the third item of${\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{2}{Cos}\;\tau_{2}} \right)},$the fourth item of${\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}\left( {R_{1}{Cos}\;\tau_{1}} \right)},$the fifth item of$\frac{\mathbb{d}}{\mathbb{d}\tau_{1}}{\left\{ {\left( {R_{1} + {\Delta\; R_{1}}} \right) - \left( {R_{2} + {\Delta\; R_{2}}} \right)} \right\}.}$

5. Calculation of the Egg Type Clothoid Using the Non-Linear Equation

When resolving the functions of F(τ₁), F′(τ₁) of the formulas 7 and 10based on the Newton-Raphson equation, it is possible to get thetangential angle τ₁, at the larger circle R₁ with respect to theparameter A, so that it is easy to get the parameter A from A²=2τ₁R₁ ².In addition, we can get a flow chart illustrating a method forcalculating the resolutions based on Newton-Raphson as follows. As showntherein, the accuracy of the calculation of the tangential angle r₁ is10⁻⁶.

FIG. 8 is a flow chart illustrating the method for calculating the eggtype clothoid parameters.

As shown therein, in a method for calculating a S shape clothoidparameter including a unknown clothoid parameter A adapted to radius(R₁, R₂) of two circles, the shortest distance between circumferentialportions of two circles and two circles, there is provided a method forcalculating an egg type clothoid parameter, comprising a step in whichan initial value of a tangential angle τ₁ is set; a step in which thevalue of (R₁−R₂−D)²−Xm² is calculated in such a manner that thetangential angle (τ₁) is compared, and when a result of the comparisonis below 0°, since it means there is not any resolution, the process isstopped, and when a result of the same is over 0°, the process iscontinued; a step in which the value of (R₁−R₂−D) ²−Xm² is compared with0, and when a result of the comparison is below 0, the tangential angleis properly adjusted, and the routine goes back to the step for settingthe initial value of the tangential value of (τ₁), and when a result ofthe same is over 0, a different fixed rate formula is set up withrespect to two circles, and one formula is formed by adding the left andright items in two formulas, for thereby calculating F′(τ₁) bydifferentiating the function F(τ₁) of the tangential angle(τ₁) and thefunction with the tangential angle (τ₁); a step in which the ratio of[G=F(τ₁)/F′(τ₁)] of two functions of is calculated; a step in which thetangential angle of τ₁=τ₁−G is calculated; and a step in which theabsolute value of the ratio (G) is compared with a permissible error(10⁻⁶), and as a result of the comparison when it is over thepermissible error, the initial value of the tangential value (τ₁) isset, and the routine is fed back to the next step of the step that thatinitial value is set, and when it is below the permissible error, thetangential angle (τ₁) is determined, and the parameter value A iscalculated using the tangential angle (τ₁).

6. Calculation Examples of Parameter A of Egg Type Clothoid

(Example 1): in the case that R₁=900, R₂=400, D=7, and

When calculating using the table: A=501.259, value verification=6.932

A result when calculating using the above algorithm: calculated in 10timesA=501.203337170923,

A result of the calculation of the verification D=7.00000000000007

Example 2 In the Case that R₁=500, R₂=120, D=2

When calculating using the diagram: A≈200

A result when calculating using the above algorithm: calculated in 7timesA=205.507481180975,

-   -   A result of calculation of verification D=2.00000000000003

In the present invention, there ar provided the drawings adapting themethods for calculating the S type and egg type parameter values A. FIG.9 is a view of a forward direction egg type design, and FIG. 10 is aview of a backward direction egg type design, and FIG. 11 is a view of aS shape egg type design.

Here, the double egg type represents that the type that two egg typesare continuously connected using an assistant circle when two circlesare crossed or distanced. FIG. 4 is a view of four types of the doubleegg type. As shown therein, there are (i)the type using an assistantcircle including two crossing circles, (ii) the type using an assistantcircle including two distanced circles, (iii) the type using anassistant circle included in two crossing circles, and (iv) the typeusing an assistant circle because the distance in the radius of twocircles is large.

FIG. 12 is a view illustrating a double egg type using an assistantcircle having two crossing circles according to the present invention,FIG. 13 is a view illustrating a double egg type using an assistantcircle having two distanced circles according to the present invention,FIG. 14 is a view illustrating a double egg type using an assistantcircuit having two crossing circles according to the present invention,FIG. 15 is a view illustrating a double egg type using an assistantcircle because a distance between the distances in the radiuses of twocircles according to the present invention, and FIG. 16 is a viewillustrating a design example of a triple egg type according to thepresent invention.

Since the double egg type represents the type that two egg types arearranged in series, it is needed to independently design two egg types.Therefore, when continuously calculating using the egg type designmethod, it is possible to achieve a design of the double egg type. Inaddition, the egg type having more than double can be designed in thesame manner as the above method.

The complex type clothoid is not well matched with the egg type. Thereare differences between the complex and egg types in that a circularportion with respect to the larger circle exists in the egg type, but itdoes not exist in the complex type(namely, the length of the circle andthe center angle of the circle are all 0). Namely, as shown in FIG. 6,there in only one difference in that KAE and KEE are placed in differentplaces but are same in the complex type.

The method for calculating the parameter A of the complex clothoid isthe same as the method for calculating the parameter A in the egg typeclothoid. When the parameter is obtained, it is possible to easy todesign the complex type clothoid. The design itself of the complex typeclothoid can directly adapt the egg type design method. In this case,the center angle of the circle with respect to the larger circle shouldbe processed with 0. Therefore, in the present invention, the method forcalculating the parameter A of the egg type clothoid can be directlyadapted to the method for calculating the parameter A in the complextype clothoid.

FIG. 17 is a view illustrating the construction of the complex typeclothoid, and FIG. 18 is a view illustrating an example of the complextype clothoid.

The method for calculating the parameter values of the S type, complextype and egg type clothoid for the design of roads according to thepresent invention are provided for the purposes of descriptions.

As described above, in the present invention, it is possible to easilycalculate the clothoid parameters A in the road designs of the S type,complex type and egg type. It is possible to achieve a fast road designfor thereby decreasing the period of road construction.

In addition, the design specification can be calculated without CAD, andthe S type and egg type road designs can be easily achieved. A desiredsimulation design can be achieved using the center coordinates of twocircles for the optimum design in the egg type.

The advantages of the present invention will be described in detail.

(i) In the present invention, it is possible to achieve a commonclothoid in one linearity and a continuous design irrespective of theegg type (forward direction, backward direction, S type, double egg typeand multiple egg type).

(ii) In the present invention, it is possible to determine the accuratespecification based on the simulation technique when the entrance andexist axes that are basic specifications and the center coordinate ofthe circle are provided as compared to the conventional art in which theconventional software programs need the accurate specification(parameter value, and the curvature and curve length). In the presentinvention, it is possible to achieve an accurate design within a fewseconds or a few minutes by adapting the algorithm according to thepresent invention as compared to the conventional art in which theconventional softwares need a few tens of minutes or a few tens ofhours. Therefore, in the present invention, it is possible tosignificantly decrease the design time period.

(iii) In addition, in the conventional softwares, only the entrance axisis given as a fixed coordinate, and it is impossible to give the existaxis a desired value, so that it is impossible to design based on theexist axis. However, in the present invention, it is possible todesignate the entrance and exist axes and to design using the designatedaxes.

(iv) In the conventional software, it is impossible to design the eggtype at one time. Namely, the egg type is classified into the separateelements such as straight line—clothoid—circularcircle—clothoid—circular curve—straight line for thereby designating adesign specification and a curving degree of a curve and straight linewith respect to each element. However, in the present invention, it ispossible to design the egg type at one time.

(v) Almost the softwares use CAD for a visual easiness. In this case,there are inconveniences for converting the file formats into the CADfiles. However, in the present invention, it is possible to achieve avisible confirmation such as enlargement, contraction, movement, etc. inthe same manner as the CAD on the screen without using the CAD program.In addition, it is possible to achieve the work directly on the CADwithout any file format conversion.

(vi) The linearity that could not be processed in the conventionalprograms can pass through a certain position, so that the presentinvention could be very advantageous when there are any limits withrespect to an obstacle, passage point, etc.

(vii) The present invention can be adapted to all types of roadlinearity designs and can be adapted irrespective of the main line andconnection roads. In addition, it has a highest advantage in thecircular curve and smoothing curve, in particular in the design of theegg type. The present invention is very useful when designing the eggtype in the connection roads.

As the present invention may be embodied in several forms withoutdeparting from the spirit or essential characteristics thereof, itshould also be understood that the above-described examples are notlimited by any of the details of the foregoing description, unlessotherwise specified, but rather should be construed broadly within itsspirit and scope as defined in the appended claims, and therefore allchanges and modifications that fall within the meets and bounds of theclaims, or equivalences of such meets and bounds are therefore intendedto be embraced by the appended claims.

1. In a method for designing a road, an improvement comprisingcalculating parameters A₁ and A₂ in an S-shaped clothoid with R₁, R₂, Dand $\frac{A_{2}}{A_{1}}$ being known values and R₁ and R₂ being radiiof two circles and A₁ and A₂ being clothoid parameters A adapted to R₁and R₂, respectively, and D being a shortest distance betweencircumferential portions of two circles, a method for calculating theS-shaped clothoid parameter comprising: a step in which an initial valueof a tangential angle τ₁ of the radius R₁ is set; a step in which thetangential angle τ₁ is compared with 0, and when a result of thecomparison is less than 0, since it means there is no solution, theprocess is stopped, and when a result of the same is greater than 0, avalue of (R₁+D+R₂)²−X_(M) ² is calculated, X_(M) being a function of anunknown value τ₁; a step in which a value of (R₁+D+R₂)²−X_(M) ² iscompared with 0, and when a result of the comparison is less than 0, τ₁is properly adjusted, and the routine goes back to the next step ofsetting the initial value of the tangential angle τ₁, and when a resultof the same is greater than 0, the function F(τ₁) is calculated and thefunction F′(τ₁) which is differentiated with τ₁ from the function F(τ₁)is calculated; a step in which a ratio$G = \frac{F\left( \tau_{1} \right)}{F^{\prime}\left( \tau_{1} \right)}$is calculated; a step in which τ₁=τ₁−G is calculated (τ₁ is reduced byG); and a step in which an absolute value of G is compared with atolerance of 10⁻⁶, and as a result of the comparison when the absolutevalue of G is greater than the tolerance, the routine is fed back to thenext step where the initial value of τ₁ is set, and when it is less thanthe tolerance, the tangential angle τ₁ is determined, and A₁, which is aclothoid parameter A adapted to R₁, is easily calculated using τ₁, andA₂, which is a clothoid parameter A adapted to R₂, is also easilycalculated using A₁ and given value $\frac{A_{2}}{A_{1}}.$
 2. The methodof claim 1, wherein in said step for calculating A₁ and A₂ withfunctions F(τ₁) and F′(τ₁, with respect to T₁, when calculations aredone for functions F(L₁) and F′(L₁) with respect to L₁, or withfunctions F(τ₂) and F′(τ₂) with respect to τ₂, or with functions F(L₂)and F′(L₂) with respect to L₂, and values of A₁ and A₂ are obtained, asame result is obtained for A₁ and A₂.
 3. The method of claim 1, whereinsaid step for calculating a solution of the functions F(τ₁) and F′(τ₁)based on a non-linear method is achieved using one selected from thegroup consisting of the Newton-Rapson equation method, bisection method,secant method, regular false method, Aitken method, successivesubstitution method, Bairstow's method, fixed point repeating method,Muller method or repeating method.
 4. The method of claim 1, whereinsaid function of F(τ₁) is as follows:${F\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{1}\left( {\tau_{1}^{2} - \frac{\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right)}} + {\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right\}} + {R_{1}{Cos}\;\tau_{1}} + {R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - \sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}$wherein t represents a constant value, and X_(M) is a function of anunknown value τ₁.
 5. The method of claim 1, wherein said function ofF′(τ₁) is as follows:${F^{\prime}\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{1}\left( {{2\tau_{1}} - \frac{4\tau_{1}^{3}}{14} + \frac{6\tau_{1}^{5}}{440} - \frac{8\tau_{1}^{7}}{25200}} \right)}} + {\frac{2}{3}{R_{2}\left( {{2t^{2}\;\tau_{1}} - \frac{4t^{4}\;\tau_{1}^{3}}{14} + \frac{6t^{6}\;\tau_{1}^{5}}{440} - \frac{8t^{8}\;\tau_{1}^{7}}{25200}} \right\}}} + {R_{1}{Sin}\;\tau_{1}} + {R_{2}t\;{{Sin}\left( {t\;\tau_{1}} \right)}} + {\frac{X_{M}}{\sqrt{\left( {R_{1} + D + R_{2}} \right)^{2} - X_{M}^{2}}}*\left\{ {{2{R_{1}\left( {1 - \frac{3\tau_{1}^{2}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + {2R_{2}t\left\{ {1 - \frac{3\left( {t\;\tau_{1}} \right)^{2}}{10} + \frac{5\left( {t\;\tau_{1}} \right)^{4}}{216} - \frac{7\left( {t\;\tau_{1}} \right)^{6}}{9360}} \right\}} - {R_{2}t\;{{Cos}\left( {t\;\tau_{1}} \right)}}} \right\}}}$where t represents a constant value, and X_(M) represents a function ofan unknown value τ₁.
 6. In a method for designing a road, an improvementcomprising calculating A in an Egg-shaped clothoid with R₁, R₂, D beingknown values and R₁ being a radius of a larger circle and R₂ being aradius of a smaller circle and D being a shortest distance betweencircumferential portions of the two circles, a method for calculating anegg-shaped clothoid parameter, comprising: a step in which an initialvalue of a tangential angle τ₁ of the radius R₁ is set; a step in whichthe tangential angle τ₁ is compared with 0, and when a result of thecomparison is less than 0, since it means there is no solution, theprocess is stopped, and when a result of the same is greater than 0, avalue of (R₁−R₂−D)²−X_(M) ² is calculated, X_(M) ² being a function ofan unknown value τ₁; a step in which the value of (R₁−R₂−D)²−X_(M) ² iscompared with 0, and when a result of the comparison is less than 0, τ₁is properly adjusted, and the routine goes back to the next step ofsetting the initial value of the tangential angle τ₁, and when a resultof the same is greater than 0, the function F(τ₁) is calculated and thefunction F′(τ₁) which is differentiated with τ₁ from the function F(τ₁)is calculated; a step in which a ratio$G = \frac{F\left( \tau_{1} \right)}{F^{\prime}\left( \tau_{1} \right)}$is calculated; a step in which τ₁=τ₁−G is calculated (τ₁ is reduced byG); and a step in which an absolute value of G is compared with atolerance of 10⁻⁶, and as a result of the comparison when the absolutevalue of G is greater than the tolerance, the routine is fed back to thenext step where the initial value of τ₁ is set, and when it is less thanthe tolerance, the tangential angle τ₁ is determined, and the value ofparameter A is easily calculated using the tangential angle τ₁.
 7. Themethod of claim 6, wherein said function F(τ₁) is as follows:${F\left( \tau_{1} \right)} = {{\frac{2}{3}R_{2}\left\{ {\left( {t\;\tau_{1}} \right)^{2} - \frac{\left( {t\;\tau_{1}} \right)^{4}}{14} + \frac{\left( {t\;\tau_{1}} \right)^{6}}{440} - \frac{\left( {t\;\tau_{1}} \right)^{8}}{25200}} \right)} - {\frac{2}{3}{R_{1}\left( \;{\tau_{1}^{2} - \frac{\;\tau_{1}^{4}}{14} + \frac{\tau_{1}^{6}}{440} - \frac{\tau_{1}^{8}}{25200}} \right\}}} + {R_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} - {R_{1}{Cos}\;\tau_{1}} + \sqrt{\left( {R_{1} + R_{2} + D} \right)^{2} - a^{2}}}$where t represents a constant value, and a represents a represents${2R_{2}\left\{ {\left( {t\;\tau_{1}} \right) - \frac{\left( {t\;\tau_{1}} \right)^{3}}{10} + \frac{\left( {t\;\tau_{1}} \right)^{5}}{216} - \frac{\left( {t\;\tau_{1}} \right)^{7}}{9360}} \right)} - {R_{2}{{Sin}\left( {t\;\tau_{1}} \right)}} - \left\{ {{2{R_{1}\left( \;{\tau_{1} - \frac{\;\tau_{1}^{3}}{10} + \frac{\tau_{1}^{5}}{216} - \frac{\tau_{1}^{7}}{9360}} \right)}} - {R_{1}{Sin}\;\tau_{1}}} \right\}$8. The method of claim 6, wherein said function of F′(τ₁) is as follows:${F^{\prime}\left( \tau_{1} \right)} = {{\frac{2}{3}{R_{2}\left( {{2{t\;}^{2}\tau_{1}} - \frac{4t^{4}\tau_{1}^{3}}{14} + \frac{6t^{6}\tau_{1}^{5}}{440} - \frac{8t^{8}\tau_{1}^{7}}{25200}} \right)}} - {\frac{2}{3}{R_{1}\left( {{2\;\tau_{1}} - \frac{4\;\tau_{1}^{3}}{14} + \frac{6\;\tau_{1}^{5}}{440} - \frac{8\;\tau_{1}^{7}}{25200}} \right\}}} + {R_{2}t\;{Sin}\;\left( {t\;\tau_{1}} \right)} + {R_{1}\;{Sin}\;\tau_{1}} - {\frac{X_{M}}{\sqrt{\left( {R_{1} + R_{2} + D} \right)^{2} - X_{M}^{2}}}*{\quad\begin{bmatrix}{{2R_{2}\left\{ {t - \frac{3t^{3}\tau_{1}^{2}}{10} + \frac{5t^{5}\tau^{4}}{216} - \frac{7t^{7}\tau_{1}^{6}}{9360}} \right\}} - {{tR}_{2}{{Cos}\left( {t\;\tau_{1}} \right)}} -} \\{{2{R_{1}\left( \;{1 - \frac{\;{3\tau_{1}^{2}}}{10} + \frac{5\tau_{1}^{4}}{216} - \frac{7\tau_{1}^{6}}{9360}} \right)}} + {R_{1}{Cos}\;\tau_{1}}}\end{bmatrix}}}}$ where X_(M)=X_(M2)−X_(M1), t represents a constantvalue, and X_(M) represents a function of an unknown value τ₁.
 9. Themethod of claim 6, wherein said step for calculating a solution of thefunctions of F(τ₁) and F′(τ₁) based on a non-linear method is achievedusing one selected from the group consisting of the Newton-Rapsonequation method, bisection method, secant method, regular false method,Aitken method, successive substitution method, Bairstow's method, fixedpoint repeating method, Muller method or repeating method.
 10. Themethod of claim 6, wherein in said step for calculating A with functionsF(τ₁) and F′(τ₁) with respect to τ₁, when A is calculated with functionsof F(τ₁), F′(τ₁) with respect to L₁, or with functions of F(τ₂), F′(τ₂)with respect to τ₂, or with functions of F(L₂), F′(L₂) with respect toL₂, and a value of A is obtained, a same result for A is obtained.